Minimizing total completion time in a two-machine flowshop: Analysis of special cases

  • Han Hoogeveen
  • Tsuyoshi Kawaguchi
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1084)


We consider the problem of minimizing total completion time in a two-machine flowshop. We present a heuristic with worst-case bound 2β/(α +β), where α and β denote the minimum and maximum processing time of all operations. Furthermore, we analyze four special cases: equal processing times on the first machine, equal processing times on the second machine, processing a job on the first machine takes time no more than its processing on the second machine, and processing a job on the first machine takes time no less than its processing on the second machine. We prove that the first special case is NP-hard in the strong sense and present an O(n log n) approximation algorithm for it with worst-case bound 4/3; we show that the other three cases are solvable in polynomial time.

1980 Mathematics Subject Classification (Revision 1991): 90B35.

Keywords and Phrases

Flowshop total completion time NP-hardness heuristics worst-case analysis special cases polynomial algorithms 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R.W. Conway, W.L. Maxwell, and L.W. Miller (1967). Theory of Scheduling, Addison-Wesley, Reading, Massachusetts.Google Scholar
  2. 2.
    M.R. Garey, D.S. Johnson, and R. Sethi (1976). The complexity of flowshop and jobshop scheduling. Mathematics of Operations Research 13, 330–348.Google Scholar
  3. 3.
    M.R. Garey and D.S. Johnson (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness, Freeman, San Francisco.Google Scholar
  4. 4.
    R.L. Graham, E.L. Lawler, J.K. Lenstra, and A.H.G. Rinnooy Kan (1979). Optimization and approximation in deterministic sequencing and scheduling: a survey. Annals of Discrete Mathematics 5, 287–326.MathSciNetGoogle Scholar
  5. 5.
    T. Gonzalez and S. Sahni (1978). Flowshop and jobshop schedules: Complexity and approximation. Operations Research 26, 36–52.Google Scholar
  6. 6.
    J.A. Hoogeveen and S.L. van De Velde (1995). Stronger Lagrangian bounds by use of slack variables: applications to machine scheduling problems. Mathematical Programming 70, 173–190.Google Scholar
  7. 7.
    E. Ignall and L. Schräge (1965). Application of the branch and bound technique for some flow-shop scheduling problems. Operations Research 13, 400–412.Google Scholar
  8. 8.
    S.L. van De Velde (1990). Minimizing the sum of the job completion times in the two-machine flow shop by Lagrangian relaxation. Annals of Operations Research 26, 257–268.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Han Hoogeveen
    • 1
  • Tsuyoshi Kawaguchi
    • 2
  1. 1.Department of Mathematics and Computing ScienceEindhoven University of TechnologyMB EindhovenThe Netherlands
  2. 2.Department of Computer Science and Intelligent SystemsOita UniversityOitaJapan

Personalised recommendations